Outline

1. Introductions
2. Preliminaries
3. Capture-recapture vs CKMR
4. History
5. CKMR Workflow
6. Expected relative reproductive output and pseudo-likelihood
7. CKMR Assumptions
8. Random (or not so random) thoughts
9. A CKMR case study: Bearded seals

Introductions

Preliminaries: Schedule

8:00 - 8:45 Close-kin mark-recapture: An overview (P. Conn)
8:45 - 9:30 An introduction to genetic data and inheritance (E. Anderson)
9:30 - 9:45 Break
9:45 - 10:30 Statistical inference for CKMR abundance estimation (P. Conn)
10:30 - 11:15 Kin finding (E. Anderson)
11:15 - 12:00 Designing a CKMR study
12:00 - 1:00 Lunch
1:00 - 5:00 R/TMB labs (full day participants only)¹

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¹ You should have followed “Setting up your computer” instructions in the workshop book!

Preliminaries: What we want you to get out of this workshop

• The basic idea of how CKMR works

• Types of CKMR models and their assumptions

• Strengths and limitations of CKMR for wildlife monitoring and management

• Basic ideas on how to design a CKMR study

• For full day participants, some ideas of how to code things up

We don’t expect anyone to be a CKMR expert after taking this workshop. There are a lot of levels of expertise required for successful CKMR implentations (including ecology, genetics, and statistics) - there are only a few people on earth that are an expert in all of these!!!

Preliminaries: Resources

Slides for morning lectures: Lecture 1 (intro) https://eriqande.github.io/tws-ckmr-2022/slides/paul-talk-1.Rmd.htm

“Book” for afternoon labs: https://eriqande.github.io/tws-ckmr-2022/

General workshop github repository: https://github.com/eriqande/tws-ckmr-2022

A CKMR website w/ more examples: https://closekin.github.io/

Mark-recapture vs. CKMR

Mark-recapture vs. CKMR

Simple MR: Lincoln-Petersen estimator

Mark-recapture vs. CKMR

CKMR

Mark-recapture vs. CKMR

CKMR

• Example: sample \(n_j = 4\) juveniles, \(M = 6\) adults (dark colored)
• Want to make inference about the number of adults
• Each juvenile has exactly two parents (\(n=8\))
• Compare genetics of sampled juveniles to sampled adults for parental relationships
• \(m=3\) parents found
• \(\hat{N} = nM/m = 8*6/3 = 16\)
• Amazing!

Mark-recapture vs. CKMR

CKMR

• Example: sample \(n_j = 4\) juveniles, \(M = 6\) adults (dark colored)
• Want to make inference about the number of adults
• Each juvenile has exactly two parents (\(n=8\))
• Compare genetics of sampled juveniles to sampled adults for parental relationships
• \(m=3\) parents found
• \(\hat{N} = nM/m = 8*6/3 = 16\)
• Amazing!

Mark-recapture vs. CKMR

Beyond Lincoln-Petersen

\(\color{red}{\rightarrow \text{Likelihood}}\)

History

• Initial development by Hans Skaug (influenced by Tore Schweder) in the late 90s
• Some convergent evolution (Rawding et al., some other cetacean work)
• In the 2010s technology improvement and further statistical development (Bravington et al. 2016) really paved the way
• Web of Science search (Oct 2022)

Figure 1: Web of Science publications on CKMR or close-kin mark-recapture.

CKMR in a nutshell

A framework for estimating adult abundance and survival using the frequency of observed kinship relationships

Parent-offspring pairs (POPs) Adult abundance and reproductive schedules (assuming age is known…)

Half-sibling pairs (HSPs) Adult abundance and survival (again assuming ages are known)

CKMR Workflow

CKMR Workflow

Pseudo-likelihood

Compare each genotyped sample to all of the others. We can then maximize the pseudo-likelihood

\(\prod_i \prod_{j>i} p_{ij}^{y_{ij}} (1-p_{ij})^{1-y_{ij}}\)

\(y_{ij}\) is a binary random variable taking on the value 1.0 if animals \(i\) and \(j\) are a match.

\(p_{ij}\) is the probability of a match

Pseudo-likelihood

Compare each genotyped sample to all of the others. We can then maximize the pseudo-likelihood

\(\prod_i \prod_{j>i} p_{ij} y_{ij} + (1-p_{ij}) (1-y_{ij})\)

\(y_{ij}\) is a binary random variable taking on the value 1.0 if animals \(i\) and \(j\) are a match.

\(p_{ij}\) is the probability of a match

\(\color{red}{\text{In reality, random variables are not independent!!}}\)

\(\color{red}{\text{So the pseudo-likelihood is an *approximation*}}\)

Expected relative reproductive output

Expected relative reproductive output

Simple example: mother-offspring pairs, knife-edged sexual maturity, no heterogeneity in reproductive success, \(b_i < b_j\)

\[\begin{equation*} p_{ij} = \begin{cases} 0, & \text{if}\ a_i(b_j) < a_{mat} \text{ or } d_i < b_j \\ 1/N_{b_j}^F, & \text{otherwise} \end{cases} \end{equation*}\]

In words: the probability of a mother-offspring pair is zero if the potential mother was reproductively immature or dead at the time of \(j\)’s birth. If the potential mother was reproductively mature, it is simply 1 over the number of reproductively mature females.

Expected relative reproductive output

Simple example: mother-offspring pairs, knife-edged sexual maturity, no heterogeneity in reproductive success, \(b_i < b_j\)

\[\begin{equation*} p_{ij} = \begin{cases} 0, & \text{if}\ a_i(b_j) < a_{mat} \\ 1/N_{b_j}^F, & \text{otherwise} \end{cases} \end{equation*}\]

In words: the probability of a mother-offspring pair is zero if the potential mother was reproductively immature at the time of \(j\)’s birth. If the potential mother was reproductively mature, it is simply 1 over the number of reproductively mature females.

\(\color{red}{\text{Ages are important!}}\)

CKMR Assumptions

1. Accurate genotyping (no false positives!)

2. Population and sampling models are accurate

3. Kinship comparisons are “independent” (or close enough…)

4. No undiagnosed relationship between kinship probabilities and sampling probabilities that can’t be explained by observed (or inferred) covariates

   • Age

   • Spatial location

   • Status (Mating hierarchy)

CKMR Assumptions - Implications

1. Accurate genotyping (no false positives!)

We need enough genetic markers to tell apart various kin groups. For parent-offspring pairs we might only need 200 SNPs or so, but for half-siblings it is nice to have 3-4K (after pruning ill-behaved loci).

\(\color{red}{\rightarrow \text{High quality tissue samples}}\)

CKMR Assumptions - Implications

2. Population model and sampling models are accurate

• For species where reproductive maturity is not instantaneous, we need to model pre-adult population dynamics, so we need some idea of early survival and reproductive schedules (decent early life history information!). We also need to get the underlying Leslie matrix right (pre vs. postbreeding census, etc.)

• Accurate sampling models have more to do with independent fates. E.g. we won’t want to model mothers and dependent offspring harvested in the same year, half-siblings harvested together, etc.

CKMR Assumptions - Implications

3. Kinship comparisons are “independent” (or close enough…)

The quality of the pseudo-likelihood as an approximation decreases as the amount of relatedness in a population increases. The usual effect when this happens in statistics is that precision (e.g., confidence intervals) is overstated.

CKMR has been conducted on populations as low as \(\approx 600\) but we don’t want to go super low.

CKMR Assumptions - Implications

So what populations is CKMR good for?

So what populations is CKMR good for?

Completed or underway as of 2022 (c/o M. Bravington)

How easy is it to conduct CKMR experiments?

• Skill level probably depends on what type of data (e.g., POP-only, POP+HSP, single cohort vs. multiple cohort)

• Relatively low cost, especially after markers and aging methods are developed (epigenetics?)

• You’re going to want to have a biologist, biometrician, and a geneticist involved. Very few people have all skills and it’s a lot to ask of a single person (especially a grad student!!!)

• Many models will need to be population- and data-dependent and will require bespoken code. That said, there are examples and templates out there that will help.

CKMR: historical ecology

CKMR: historical ecology

CKMR: historical ecology

A CKMR case study: bearded seals

Paper in prep (Taras, Conn, Quakenbush, Bravington, Baylis). Annual sampling of bearded seal subsistence harvests (tissue samples + teeth) by ADF&G

A CKMR case study: Bearded

Our paper isn’t published yet so I can’t make all data public yet. But we can still use life history information, approximate sample sizes, and kin finding data to help motivate ideas and conduct some modeling exercises.

• Size of Beringia DPS thought to be 400-500K seals!

• We had something like 2000 tissue samples over \(\approx 20\) years.

• Can we use this to get an idea of overall abundance? What special things will we need to consider when fitting models to these data?